Monday, December 31, 2007

Fireside Problem

Over the Christmas period I have been grappling with a problem that has been hanging around with me for some time.
Now, it is possible to generate highly disordered sequences of 1s and 0s using an algorithm. An algorithm is also expressible as a sequence of 1s and 0s. However, whereas disordered sequences may be long – say of length Lr, the length of the algorithm sequence generating it, Lp, may be short. That is Lr is less than Lp. The number of possible algorithm sequences will be 2^Lr. The number of possible disordered sequences will be nearly equal to 2^Lp. So, because 2^Lp is much less than 2^Lr, then it is clear that short algorithms can only generate a very limited number of all the possible disordered sequences. For me, this result has a counter intuitive consequence: It suggests that there is a small class of disordered sequences that have a special status - namely the class of disordered sequences that can be generated algorithmically. But why should a particular class of disordered sequences be so mathematically favoured when in one sense every disordered sequence is like every other disordered sequence in that they all have the same statistical profile? My intuitions suggest that all disordered sequences are in some sense mathematically equal, and yet it seems that algorithms confer a special status on a small class of these sequences.
I think I now know where the answer to this intuitive contradiction lies. To answer it we have to go back to the network view of algorithmic change. If we take a computer like a Turing machine then it seems that it wires the class of all binary sequences into a network in a particular way, and it is the bias introduced by the network wiring that leads to certain disordered configurations being apparently favored. It is possible to wire the network together in other ways that would favour another class of disordered sequences. In short the determining variable isn’t the algorithm, but the network wring, which is a function of the computing model being used. It is the computing model inherent in the network wiring that has as much degree of freedom as does a sequence as long as Lr. Thus, in as much as any disordered sequence can have an algorithmically favoured position depending on the network wiring used by the computing model, then in that sense no disordered sequence is absolutely favoured over any other.

Well, that’ll have to do for now; I suppose I had better get back to the Intelligent Design Contention.

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